DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Query-register projectors and mismatch indicators

Definition

For the address register of Boolean oracle query model for search lower bounds, let $Π_{j}$ be the orthogonal projection onto address $j$ , tensored with the identity on the response and work registers, for $0 \leq j < N$ . For any finite input domain $D \subseteq {0, 1}^{N}$ , define the mismatch matrix $Δ_{j}$ by $(Δ_{j})_{x, y} = 1$ when $x_{j + 1} \neq y_{j + 1}$ and $0$ otherwise, for $x, y \in D$ . This model has no separate skip-query address; if an implementation adjoins one, its basis projection is included as an additional orthogonal summand.

Facts & Assumptions

Given: $N \geq 1$ , the finite address register with ordered computational basis, and the response/work tensor factors.

[F1]

Boolean oracle query model for search lower bounds: the address basis is indexed by $0, \dots, N - 1$ and is orthonormal.

[F2]

The computational basis is orthonormal: the declared register basis is an orthonormal basis.

[F3]

Dirac Notation Conversion: $∣ i ⟩ ⟨ i ∣$ is the rank-one map $v \mapsto ⟨ v, i ⟩ i$ in the root inner-product convention.

Verification

1.1givenF1F2F3algebra

For each address $j$ , $Π_{j} = ∣ j ⟩ ⟨ j ∣ \otimes I$ is self-adjoint and idempotent by [F1]–[F3], so it is an orthogonal projection. If $j \neq k$ , orthogonality gives $Π_{j} Π_{k} = 0$ .

2.1F1F2F3step 1.1algebra∎

The address vectors form a complete basis by [F1]–[F2], hence $\sum_{j = 0}^{N - 1} ∣ j ⟩ ⟨ j ∣ = I$ and therefore $\sum_{j = 0}^{N - 1} Π_{j} = I$ on address, response, and work registers. If a no-query basis state is adjoined, the same basis expansion includes its projection and the search-address sum is then at most $I$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani, Strengths and Weaknesses of Quantum Computing (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (2025) (standard reference, not scraped)